Solve for x: x/(10 + x) = 0.97.

Step-by-step solution

1. 1

   Read the equation as a share of a total. $x$ out of $10+x$ must be $97\%$, so the fixed $10$ accounts for the remaining $3\%$. That immediately gives the total as roughly $\tfrac{10}{0.03}\approx333$, and $x$ as about $323$.

2. 2

   Convert to a fraction and note the domain.

   $0.97=\frac{97}{100}\qquad\Longrightarrow\qquad \frac{x}{10+x}=\frac{97}{100},\qquad x\neq-10$

   Since $97$ is prime, this fraction is already in lowest terms — unlike $0.98=\tfrac{49}{50}$, which reduces.

3. 3

   Cross-multiply.

   $100x=97(10+x)$

4. 4

   Expand and collect.

   $100x=970+97x\qquad\Longrightarrow\qquad 3x=970$

   The coefficient $3=100-97$ is exactly the missing three percentage points, expressed in hundredths.

5. 5

   Divide, and note that the decimal repeats.

   $x=\frac{970}{3}=323.\overline{3}\approx323.33$

   Since $3$ does not divide $970$ and is not built from $2$s and $5$s, the exact fraction is the better answer. This is the structural difference from the $0.98$ case, where $100-98=2$ divides evenly and gives the whole number $490$.

6. 6

   Verify with exact arithmetic.

   $\frac{970/3}{10+970/3}=\frac{970/3}{1000/3}=\frac{970}{1000}=0.97\ \checkmark$

   and the closed form $x=\dfrac{ap}{1-p}=\dfrac{10(0.97)}{0.03}=\dfrac{9.7}{0.03}=323.\overline{3}$ agrees ✓.